Results on the Guarded Fragment with Equivalence or Transitive Relations

Abstract

We study the problem of the satisfiability of guarded formulas in models in which some distinguished binary symbols are interpreted as equivalence relations or as transitive relations. We sharpen the undecidability result for the two-variable guarded fragment with transitive relations by reducing the number of transitive relations to two. We prove that the satisfiability problem for the two-variable guarded fragment with two equivalence relations is 2EXPTIME-complete. We consider the guarded fragment with equivalence relations in guards and show that this variant is easily reducible to the variant with transitive relations in guards. However, in the case of two variables, the version with equivalence relations is easier: NEXPTIME-complete. Finally we show that the decidability results for the guarded fragment with either equivalence relations or transitive relations in guards cannot be generalized to the loosely guarded fragment.